John B. Fraleigh's A First Course in Abstract Algebra - Instructor's Solutions PDF

John B. Fraleigh's A First Course in Abstract Algebra - Instructor's Solutions PDF

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March 16, 2018
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By John B. Fraleigh

ISBN-10: 0201763907

ISBN-13: 9780201763904

Instructor's suggestions guide to a primary direction in summary Algebra that's uploaded the following: http://bibliotik.org/torrents/27184 comprises solutions to all problems.

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The even permutations in S3 are ρ0 = (12)(12), ρ1 = (1, 2, 3) = (1, 3)(1, 2), and ρ2 = (1, 3, 2) = (1, 2)(1, 3). ρ0 ρ1 ρ2 ρ0 ρ0 ρ1 ρ2 ρ1 ρ1 ρ2 ρ0 ρ2 ρ2 ρ0 ρ1 25. Viewing a permutation σ in Sn as permuting the rows of the identity matrix In , we see that if σ could be expressed as both an even and odd number of transpositions (giving row interchanges), then the matrix resulting from applying σ to In would have both determinant 1 and determinant -1. 26. If σ is a permutation and τ = (i, j) is a transposition in Sn , then by considering whether i and j are in the same or different orbits of σ, we can show that the number of orbits of σ and of τ σ differ by 1.

Suppose that G contains two elements, a and b, of order 2. Then (ab)2 = abab = aabb = ee = e and ab = e because the inverse of a is a itself. Thus ab also has order 2. It is easily checked that then {e, a, b, ab} is a subgroup of G of order 4. But this is impossible because n is odd and 4 does not divide 2n. Thus there can’t be two elements of order 2. 37. Let G be of order ≥ 2 but with no proper nontrivial subgroups. Let a ∈ G, a = e. Then a is a nontrivial subgroup of G, and thus must be G itself.

45. We can work with Zn . Let d divide n. Then n/d = {0, n/d, 2n/d, · · · , (d − 1)n/d} is a subgroup of Zn of order d. It consists precisely of all elements x ∈ Zn such that dx = x + x + · · · + x for d summands is equal to 0. Because an element x of any subgroup of order d of Zn must satisfy dx = 0, we see that n/d is the only such subgroup. Because the order of a subgroup must divide the order of the whole group, we see that these are the only subgroups that Zn has. 46. 16. By the preceding exercise, there is a unique such subgroup of order d dividing n.